Fitting statistical models in bivariate allometry

Several attempts have been made in recent years to formulate a general explanation for what appear to be recurring patterns of allometric variation in morphology, physiology, and ecology of both plants and animals (e.g. the Metabolic Theory of Ecology, the Allometric Cascade, the Metabolic‐Level Boundaries hypothesis). However, published estimates for parameters in allometric equations often are inaccurate, owing to undetected bias introduced by the traditional method for fitting lines to empirical data. The traditional method entails fitting a straight line to logarithmic transformations of the original data and then back‐transforming the resulting equation to the arithmetic scale. Because of fundamental changes in distributions attending transformation of predictor and response variables, the traditional practice may cause influential outliers to go undetected, and it may result in an underparameterized model being fitted to the data. Also, substantial bias may be introduced by the insidious rotational distortion that accompanies regression analyses performed on logarithms. Consequently, the aforementioned patterns of allometric variation may be illusions, and the theoretical explanations may be wide of the mark. Problems attending the traditional procedure can be largely avoided in future research simply by performing preliminary analyses on arithmetic values and by validating fitted equations in the arithmetic domain. The goal of most allometric research is to characterize relationships between biological variables and body size, and this is done most effectively with data expressed in the units of measurement. Back‐transforming from a straight line fitted to logarithms is not a generally reliable way to estimate an allometric equation in the original scale.     

Allometric equations for predicting body mass of dinosaurs

We use data from the literature to compare two statistical procedures for estimating mass (or size) of quadrupedal dinosaurs and other extraordinarily large animals in extinct lineages. Both methods entail extrapolation from allometric equations fitted to data for a reference group of contemporary animals having a body form similar to that of the dinosaurs. The first method is the familiar one of fitting a straight line to logarithmic transformations, followed by back-transformation of the resulting equation to a two-parameter power function in the arithmetic scale. The second procedure entails fitting a two-parameter power function directly to arithmetic data for the extant forms by nonlinear regression. In the example presented here, the summed circumferences for humerus plus femur for 33 species of quadrupedal mammals was the predictor variable in the reference sample and body mass was the response variable. The allometric equation obtained by back-transformation from logarithms was not a good fit to the largest species in the reference sample and presumably led to grossly inaccurate estimates for body mass of several large dinosaurs. In contrast, the allometric equation obtained by nonlinear regression described data in the reference sample quite well, and it presumably resulted in better estimates for body mass of the dinosaurs. The problem with the traditional analysis can be traced to change in the relationship between predictor and response variables attending transformation, thereby causing measurements for large animals not to be weighted appropriately in fitting models by least squares regression. Extrapolations from statistical models obtained by back-transformation from lines fitted to logarithms are unlikely to yield reliable predictions for body size in extinct animals. Numerous reports on the biology of dinosaurs, including recent studies of growth, may need to be reconsidered in light of our findings.     

A comparison of methods for fitting allometric equations to field metabolic rates of animals

We re-examined data for field metabolic rates of varanid lizards and marsupial mammals to illustrate how different procedures for fitting the allometric equation can lead to very different estimates for the allometric coefficient and exponent. A two-parameter power function was obtained in each case by the traditional method of back-transformation from a straight line fitted to logarithms of the data. Another two-parameter power function was then generated for each data-set by non-linear regression on values in the original arithmetic scale. Allometric equations obtained by non-linear regression described the metabolic rates of all animals in the samples. Equations estimated by back-transformation from logarithms, on the other hand, described the metabolic rates of small species but not large ones. Thus, allometric equations estimated in the traditional way for field metabolic rates of varanids and marsupials do not have general importance because they do not characterize rates for species spanning the full range in body size. Logarithmic transformation of predictor and response variables creates new distributions that may enable investigators to perform statistical analyses in compliance with assumptions underlying the tests. However, statistical models fitted to transformations should not be used to estimate parameters of equations in the arithmetic domain because such equations may be seriously biased and misleading. Allometric analyses should be performed on values expressed in the original scale, if possible, because this is the scale of interest.     

On the use of log‐transformation versus nonlinear regression for analyzing biological power laws

Xiao and colleagues re‐examined 471 datasets from the literature in a major study comparing two common procedures for fitting the allometric equation y = ax^b to bivariate data (Xiao et al., 2011). One of the procedures was the traditional allometric method, whereby the model for a straight line fitted to logarithmic transformations of the original data is back‐transformed to form a two‐parameter power function with multiplicative, lognormal, heteroscedastic error on the arithmetic scale. The other procedure was standard nonlinear regression, whereby a two‐parameter power function with additive, normal, homoscedastic error is fitted directly to untransformed data by nonlinear least squares. Xiao and colleagues articulated a simple (but explicit) protocol for fitting and comparing the alternative models, and then used the protocol to examine each of the datasets in their compilation. The traditional method was said to provide a better fit in 69% of the cases and an equivalent fit in another 15%, so the investigation appeared to validate findings from a large majority of prior studies on allometric variation. However, focus for the investigation by Xiao and colleagues was overly narrow, and statistical models apparently were not validated graphically in the scale of measurement. The present study re‐examined a subset of the cases using a larger pool of candidate models and graphical validation, and discovered complexities that were overlooked in their investigation. Some datasets that appeared to be described better by the traditional method actually were unsuited for use in an allometric analysis, whereas other datasets were not described adequately by a two‐parameter power function, regardless of how the model was fitted. Thus, conclusions reached by Xiao and colleagues are not well supported and their paradigm for fitting allometric equations is unreliable. Future investigations of allometric variation should adopt a more holistic approach and incorporate graphical validation on the original arithmetic scale. © 2014 The Linnean Society of London, Biological Journal of the Linnean Society, 2014, 113 , 1167–1178.     

Bias in interspecific allometry: examples from morphological scaling in varanid lizards

The traditional approach to allometric analysis entails the fitting of a straight line to logarithmic transformations of the data, after which parameters in a two-parameter allometric equation are estimated by back-transformation to the original scale. We re-examined published data for dimensions of the limbs in 22 species of varanid lizards to illustrate the biases that can be introduced into allometric analyses by applying the aforementioned protocol. Statistical models fit to the original data by linear and nonlinear regression conformed better with underlying assumptions than did models obtained by back-transformation from logarithms, and the former generally were better than the latter for describing limb dimensions over the full range in body size. Allometric exponents estimated by the traditional method therefore were based on inappropriate and inaccurate statistical models and, consequently, were biased and misleading. Investigators can avoid problems such as these by performing preliminary graphical and statistical analyses on data in their original scale and by validating the fitted model. Logarithmic transformations should be used sparingly and only for cause.  © 2009 The Linnean Society of London, Biological Journal of the Linnean Society , 2009, 96 , 296–305.     

Unanticipated consequences of logarithmic transformation in bivariate allometry

Parameters in the two-parameter allometric equation are commonly estimated by fitting a straight line to logarithmic transformations of the original data and by back-transforming the resulting model to the arithmetic scale. However, log transformation distorts the relationship between the predictor and response variables, and this distortion may be sufficient to lead unsuspecting investigators to analyze data that actually are unsuited for allometric research. Two data sets from the current literature are re-examined here to illustrate instances in which log transformation caused ugly data to look deceptively good. One of the investigations focused on the scaling of metabolism to body mass in evolutionary transitions from prokaryotic to protistan to metazoan levels of organization whereas the other addressed the scaling of intestines to body size in rodents. In both instances investigators were led to conclusions that are not supported by the original data. Problems of the sort described here can readily be avoided simply by performing preliminary graphical analysis of observations expressed in the original units and by validating the final model in the arithmetic domain.     

Regression equations between body and head measurements in the broad-snouted Caiman (Caiman latirostris)

In the present study, regression equations between body and head length measurements for the broad-snouted caiman (Caiman latirostris) are presented. Age and sex are discussed as sources of variation for allometric models. Four body-length, fourteen head-length, and ten ratio variables were taken from wild and captive animals. With the exception of body mass, log-transformation did not improve the regression equations. Besides helping to estimate body-size from head dimensions, the regression equations stressed skull shape changes during the ontogenetic process. All age-dependent variables are also size-dependent (and consequently dependent on growth rate), which is possibly related to the difficulty in predicting age of crocodilians based on single variable growth curves. Sexual dimorphism was detected in the allometric growth of cranium but not in the mandible, which may be evolutionarily related to the visual recognition of gender when individuals exhibit only the top of their heads above the surface of the water, a usual crocodilian behavior.     

Modeling the influence of body size on weightlifting and powerlifting performance

The purpose of this study was to examine 1) if lifting performance in both the weightlifting (WL) and powerlifting (PL) scale with body mass (M) in line with theory of geometric similarity, and 2) whether there are any gender differences in the allometric relationship between lifting performance and body size. This was performed by analyzing ten best WL and PL total results for each weight class, except for super heavyweight, achieved during 2000-2003. Data were analysed with the allometric and second-order polynomial model, and detailed regression diagnostics was applied in order to examine appropriateness of the models used. Results of the data analyses indicate that 1) women's WL and men's PL scale for M in line with theory of geometric similarity, 2) both WL and PL mass exponents are gender-specific, probably due to gender differences in body composition, 3) WL and PL results scale differently for M possibly due to their structural and functional differences. However, the obtained mass exponents does not provide size-independent indices of lifting performances since the allometric model exhibit a favourable bias toward middleweight lifters in most lifting data analyzed. Due to possible deviations from presumption of geometric similarity among lifters, future studies on scaling lifting performance should use fat-free mass and height as indices of body size.     

Allometry, antilog transformations, and the perils of prediction on the original scale

Biologists often use allometric equations that take the form of power functions (e.g., Y = aM(b), where M stands for mass and a and b are empirically fitted constants). Typically, these allometric equations are fitted by taking the antilog of log-log regressions. Predictions from these allometric equations are biased, and the bias my be appreciable. Methods for making predictions that correct for the bias are available, but they have rarely, if ever, been used by ecological and evolutionary physiologists. Just as physiologists would not use an instrument that was not properly calibrated, they should not use allometric equations to make predictions unless they account for the bias of those predictions. We analyzed 20 interspecific and 10 intraspecific data sets. We compared predictions from standard allometric equations with those from several alternative methods. Our analyses suggest that the bias of predictions from interspecific data sets may be substantial. For the intraspecific data sets we analyzed, the bias was likely to be small. Biologists, including ecological and evolutionary physiologists, should exercise care when using allometric equations to make predictions, particularly given that methods to adjust for bias are easily implemented.     

Analysis of the independent power of age-related,anthropometric and mechanical factors as determinants of the structure of radius and tibia in normal adults. A pQCT study

To compare the independent influence of mechanical and non-mechanical factors on bone features, multiple regression analyses were performed between pQCT indicators of radius and tibia bone mass, mineralization, design and strength as determined variables, and age or time since menopause (TMP), body mass, bone length and regional muscles’ areas as selected determinant factors, in Caucasian, physically active, untrained healthy men and pre- and post-menopausal women. In men and pre-menopausal women, the strongest influences were exerted by muscle area on radial features and by both muscle area and bone length on the tibia. Only for women, was body mass a significant factor for tibia traits. In men and pre-menopausal women, mass/design/strength indicators depended more strongly on the selected determinants than the cortical vBMD did (p<0.01-0.001 vs n.s.), regardless of age. However, TMP was an additional factor for both bones (p<0.01-0.001). The selected mechanical factors (muscle size, bone lengths) were more relevant than age/TMP or body weight to the development of allometrically-related bone properties (mass/design/strength), yet not to bone tissue “quality” (cortical vBMD), suggesting a determinant, rather than determined role for cortical stiffness. While the mechanical impacts of muscles and bone levers on bone structure were comparable in men and pre-menopausal women, TMP exerted a stronger impact than allometric or mechanical factors on bone properties, including cortical vBMD.     

Taxonomic versus allometric constraints on non‐linear interaction strengths

Recently, the importance of body mass and allometric scaling for the structure and dynamics of ecological networks has been highlighted in several ground‐breaking studies. However, advances in the understanding of generalities across ecosystem types are impeded to a considerable extent by a methodological dichotomy contrasting a considerable portion of marine ecology on the one hand opposite to traditional community ecology on the other hand. Many marine ecologists are bound to the taxonomy‐neglecting size spectrum approach when describing and analysing community patterns. In contrast, the mindset of the other school is focused on taxonomies according to the Linnean system at the cost of obscuring information due to applying species or population averages of body masses and other traits. Following other pioneering studies, we addressed this lingering gap, and studied non‐linear interaction strengths (i.e. functional responses) between two taxonomically‐distinct terrestrial arthropod predators (centipedes and spiders) of varying individual body masses and their prey. We fitted three non‐linear functional response models to the data: (1) a taxonomic model not accounting for variance in body masses amongst predator individuals, (2) an allometric model ignoring taxonomic differences between predator individuals, and (3) a combined model including body mass and taxonomic effects. Ranked according to their AICs, the combined model performs better than the allometric model, which provides a superior fit to the data than the taxonomic model. These results strongly indicate that the body masses of predator and prey individuals were responsible for most of the variation in non‐linear interaction strengths. Taxonomy explained some specific patterns in allometric exponents between groups and revealed mechanistic insights in predation efficiencies. Reconciling quantitative allometric models as employed by the marine size‐spectrum approach with taxonomic information may thus yield quantitative results that are generalized across ecosystem types and taxonomic groups. Using these quantitative models as novel null models should also strengthen subsequent taxonomic analyses.     

Logarithmic transformation bias in allometry

It has been known for some time (DJ Finney, J. Roy. Stat. Soc. Suppl. 7:155–161, 1941) that transformation of an arithmetic data set to logarithms results in biased estimates when predicted values from a leastsquares regression are detransformed back to arithmetic units. Predicted values are estimates of the geometric mean of the dependent variable at that value of the independent variable, rather than the arithmetic mean. Since the geometric mean is always less than the arithmetic mean, detransformed predictions will underestimate the value in question. This bias affects the interpretations of allometric equations used for estimation, such as predicting fossil body mass from skeletal dimensions, and applications of allometry as a “criterion of subtraction,” in which residual variation is evaluated. A number of parametric and nonparametric corrections for transformation bias have been developed. Although this problem is relatively unexplored in mammalian morphometrics, it has received considerable attention in other disciplines that use power functions structurally identical to the allometric equation. Insights into transformation bias and the use of correction terms from economics, limnology, forestry, and hydrology are reviewed and interpreted for application to mammalian allometry. © 1993 Wiley-Liss, Inc.     

Mathematical Model for the Contribution of Individual Organs to Non-Zero Y-Intercepts in Single and Multi-Compartment Linear Models of Whole-Body Energy Expenditure

Mathematical models for the dependence of energy expenditure (EE) on body mass and composition are essential tools in metabolic phenotyping. EE scales over broad ranges of body mass as a non-linear allometric function. When considered within restricted ranges of body mass, however, allometric EE curves exhibit ‘local linearity.’ Indeed, modern EE analysis makes extensive use of linear models. Such models typically involve one or two body mass compartments (e.g., fat free mass and fat mass). Importantly, linear EE models typically involve a non-zero (usually positive) y-intercept term of uncertain origin, a recurring theme in discussions of EE analysis and a source of confounding in traditional ratio-based EE normalization. Emerging linear model approaches quantify whole-body resting EE (REE) in terms of individual organ masses (e.g., liver, kidneys, heart, brain). Proponents of individual organ REE modeling hypothesize that multi-organ linear models may eliminate non-zero y-intercepts. This could have advantages in adjusting REE for body mass and composition. Studies reveal that individual organ REE is an allometric function of total body mass. I exploit first-order Taylor linearization of individual organ REEs to model the manner in which individual organs contribute to whole-body REE and to the non-zero y-intercept in linear REE models. The model predicts that REE analysis at the individual organ-tissue level will not eliminate intercept terms. I demonstrate that the parameters of a linear EE equation can be transformed into the parameters of the underlying ‘latent’ allometric equation. This permits estimates of the allometric scaling of EE in a diverse variety of physiological states that are not represented in the allometric EE literature but are well represented by published linear EE analyses.     

Adjusting powerlifting performances for differences in body mass

It has been established that, in the sports of Olympic weightlifting (OL) and powerlifting (PL), the relationship between lifting performance and body mass is not linear. This relationship has been frequently studied in OL, but the literature on PL is less extensive. In this study, PL performance and body mass, for both men and women, was examined by using data from the International Powerlifting Federation World Championships during 1995-2004. Nonlinear regression was used to apply 7 models (including allometric, polynomial, and power models) to the data. The results of this study indicate that the relationship between PL performance and body mass can be best modeled by the equation y = a - bx(-c), where y is the weight lifted (in kg) in the squat, bench press, or deadlift, x is the body mass of the lifter (in kg), and a, b, and c are constants. The constants a, b, and c are determined by the type of lift (squat, bench press, or deadlift) and the gender of the lifter and were obtained from the regression analysis. Inspection of the plots of raw residuals (actual performance minus predicted performance) vs. body mass revealed no body mass bias to this formula in contrast to research into other handicapping formulas. This study supports previous research that found a bias toward lifters in the intermediate weight categories in allometric fits to PL data.     

Bioelectrical impedance estimation of fat-free body mass in children and youth: a cross-validation study.

The purposes of this study were to develop and cross-validate the "best" prediction equations for estimating fat-free body mass (FFB) from bioelectrical impedance in children and youth. Predictor variables included height2/resistance (RI) and RI with anthropometric data. FFB was determined from body density (underwater weighing) and body water (deuterium dilution) (FFB-DW) and from age-corrected density equations, which account for variations in FFB water and bone content. Prediction equations were developed using multiple regression analyses in the validation sample (n = 94) and cross-validated in three other samples (n = 131). R2 and standard error of the estimate (SEE) values ranged from 0.80 to 0.95 and 1.3 to 3.7 kg, respectively. The four samples were then combined to develop a recommended equation for estimating FFB from three regression models. R2 and SEE values and coefficients of variation from these regression equations ranged from 0.91 to 0.95, 2.1 to 2.9 kg, and 5.1 to 7.0%, respectively. As a result of all cross-validation analyses, we recommend the equation FFB-DW = 0.61 RI + 0.25 body weight + 1.31, with a SEE of 2.1 kg and adjusted R2 of 0.95. This study demonstrated that RI with body weight can predict FFB with good accuracy in Whites 10-19 yr old.     

Mechanics of posture and gait of some large dinosaurs

Dimensions of dinosaur bones and of models of dinosaurs have been used as the basis for calculations designed to throw light on the posture and gaits of dinosaurs.
Estimates of the masses of some dinosaurs, obtained from the volumes of models, are compared with previous estimates. The positions of dinosaurs' centres of mass, derived from models, show that some large quadrupedal dinosaurs supported most of their weight on their hind legs and were probably capable of rearing up on their hind legs.
Distributions of bending moments along the backs of large dinosaurs are derived from measurements on models. The tensions required in epaxial muscles to enable Diplodocus to stand are calculated. It is likely that the long neck of this dinosaur was supported by some structure running through the notches in the neural spines of its cervical and dorsal vertebrae. The nature of this hypothetical structure is discussed.
An attempt is made to reconstruct the walking gait of sauropod dinosaurs, from the pattern of footprints in fossil tracks.
The dimensions of dinosaur leg bones are compared to predictions for mammals of equal body mass, obtained by extrapolation of allometric equations. Their dimensions are also used to calculate a quantity which is used as an indicator of strength in bending. Comparisons with values for modern animals lead to speculations about the athletic performance of dinosaurs.
Estimates of pressures exerted on the ground by the feet of dinosaurs are used in a discussion of the ability of dinosaurs to walk over soft ground.     

Gestation length, metabolic rate, and body and brain weights in primates: epigenetic effects

The relationship of brain and body weights can be expressed in log-log regression: log (brain weight) = log (A) + B log (body weight). To investigate further the weights' similarity, gestation length and brain and body weights were determined from the literature for 46 primate genera. The results of allometric and path regression analyses suggest that the relationship between brain and body weights may not be mainly pleiotropic in the order Primates. The correlation between brain and body weights appears to be due to epigenetic factors in hyperplastic growth related to time constraint by gestation length and to energy utilization limitations imposed by metabolic rate.     

The ontogenetic allometry of body morphology and chemical composition in dairy goat wethers

We studied the ontogenetic growth of goat wethers (castrated male goats) of the Saanen and Swiss Alpine breeds based on a large range of intraspecific body mass (BM). The body parts and the chemical constituents of the empty body were described by the allometric function by using BM and the empty body mass (EBM) as the predictors for morphological traits and chemical composition, respectively. We fitted the allometric scaling function by applying the SAS NLMIXED procedure, but to evaluate assumptions regarding variances in morphological and compositional traits, we combined the scaling function with homoscedastic (MOD1), and the heteroscedastic exponential (MOD2) and power-of-the-mean (MOD3) variance functions. We also predicted the ontogenetic growth by using the traditional log-log transformation and back-transformed results into the arithmetic scale (MOD4). We obtained predictions from MOD4 in the arithmetic scale by a two-step process, and evaluated MOD1, MOD2 and MOD3 by a model selection framework, and compared MOD4 with MOD1, MOD2 and MOD3 based on goodness-of-fit measures. Based on information criteria for model selection, heterogeneous variance functions were more likely to describe 10 over 36 traits with a low level of model selection uncertainty. One trait was predicted by averaging the MOD1 and MOD2 variance functions; and nine traits were better described by averaging the MOD2 and MOD3 variance functions. The predictions for other 16 traits were averaged from MOD1, MOD2 and MOD3. However, MOD4 better described 11 traits according to the goodness-of-fit measures. Depending on the variable being analyzed, the body parts and the chemical amounts exhibited the three types of allometric behavior with respect to BM and EBM, that is, positive, negative and isometric ontogenetic growth. Reference BMs, that is, 20, 27, 35 and 45 kg, were used to compute the net protein and energy requirements based on the first derivative of the scaling function, and the results were presented in reference to the EBM and EBM^0.75. Both the net protein and energy requirements scaled to EBM^0.75 increased from 20 to 45 kg of BM.     

基于树木起源、立地分级和龄组的单木生物量模型

以马尾松(Pinus massoniana)和落叶松(Larix)的大样本实测资料为建模样本,以独立抽取的样本为验证样本,把样本按起源、立地和龄组进行分级,采用与材积相容的两种相对生长方程,分普通最小二乘和两种加权最小二乘,对地上部分总生物量、地上各部分生物量和地下生物量进行模型拟合和验证,使用决定系数、均方根误差、总相对误差和估计精度等8项统计量对结果进行分析。结果表明:两个树种地上部分总生物量,立地分类方法,模型的拟合结果和适用性都最优;马尾松VAR模型较优,而落叶松CAR模型较好;两种加权最小二乘方法,在建模样本和验证样本中表现得不一致。在建模样本中,加权回归2(权重函数1/f^0.5)略优于加权回归1(权重函数1/y^0.5),但在验证样本中,加权回归1却明显优于加权回归2。而同时满足建模样本拟合结果最优和验证样本检验结果最优的组合中,只有加权回归1。两个树种地上部分各分量生物量,模型拟合结果和适用性,均为干材最优,树叶最差、树枝和树皮居中,样本分类、模型类型和加权最小二乘方法对干材生物量的影响,规律和地上部分总生物量相同;样本分类、模型类型和加权最小二乘方法的最优组合,用验证样本检验的结果,总相对误差树枝不超过±10.0%,树皮不超过±5.0%,树叶马尾松不超过±30.0%,落叶松不超过±20.0%。两个树种地下部分(根)生物量,样本按龄组分类方法,模型拟合结果最优,与材积相容的模型总体上优于与地上部分总生物量相容模型。